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5-6: The Law of Sines

If we have a non right
triangle, we cannot use the
primary trig ratios, so we
must explore new
trigonometric relationships.

One such relationship is
called the Sine Law which
states the following:
C

B
A
a
b
c


sin A sin B sin C
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Law of Sines: Solve for Sides
Have: two angles, one
side opposite one of the
given angles
Solve for: missing side
opposite the other given
angle
A
b
c
C
a
B
Missing Side
a
b
=
sin A
sin B
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Law of Sines: Solve for Angles
Have: two sides and one
of the opposite angles
Solve for: missing angle
opposite the other given
angle
Missing Angle
A
b
c
C
a
B
a
b
=
sin A
sin B
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Examples Sine Law

We can use these new trigonometric relationships in
solving for unknown sides and angles in acute triangles:

ex 4. Find A in ABC if a = 10.4, c = 12.8 and C = 75°

ex 5. Find a in ABC if A = 84°, B = 36°, and b = 3.9

ex 6. Solve EFG if E = 82°, e = 11.8, and F = 25°

There is one limitation on the Sine Law, in that it can
only be applied if a side and its opposite angle is known.
If not, the Sine Law cannot be used.
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Examples Sine Law

Mark is a landscaper who is
creating a triangular
planting garden. The
homeowner wants the
garden to have two equal
sides and contain an angle
of 75°. Also, the longest
side of the garden must be
exactly 5 m.


(a) How long is the plastic
edging that Mark needs to
surround the garden?
(b) Determine the area of
the garden.
F
75
5 meters
G
H
5
Area Formula:
A
b
Have:
c
two sides, included angle
C
1
K=
a b sin
C
t
2
two sides
a
B
included angle
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Example Area Formula

Find the area of triangle ABC if b=21.2,
c=16.5, and A=25 degrees.
Cosine Law

The Cosine Law is stated
the following:

a² = b² + c² - 2bcCosA
C

B
A

We can use the Cosine Law
to work in triangles in which
the Sine Law does not work
- triangles in which we know
all three sides and one in
which we know two sides
plus the contained angle.
C
A
B
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Law of Cosines: Solve for Side
A
b
Have: two sides,
included angle
Solve for: missing side
2
c
=
2
a
+
C
2
b
c
a
B
– 2 a b cos C
(missing side)2 = (one side)2 + (other side)2 – 2 (one side)(other side) cos(included angle)
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Law of Cosines: Solve for Missing
Angle
A
Have: three sides
b
c
Solve for: missing angle
C
a
B
Side Opposite
Missing Angle
Missing Angle
a2 + b2 – c2
cos C =
2ab
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Examples Cosine Law

We can use these new trigonometric relationships in
solving for unknown sides and angles in acute triangles:

ex 7. Find c in CDE if C = 56°, d = 4.7 and e = 8.5

ex 8. Find G in GHJ if h = 5.9, g = 9.2 and j = 8.1

ex 9. Solve ΔCDE if D = 49°, e = 3.7 and c = 5.1
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Examples Cosine Law

A boat travels 60 km due east. It then adjusts its course by
25°northward and travels another 90km in this new direction.
How far is the boat from its initial position to the nearest
kilometre?
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(G) Links

For help with right triangle trig:
http://id.mind.net/~zona/mmts/trigonometryRealms/introductio
n/rightTriangle/trigRightTriangle.html

For help with the Sine Law
http://www.themathpage.com/aTrig/law-of-sines.htm



For help with the Cosine Law
http://www.themathpage.com/aTrig/law-of-cosines.htm
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